AP Calc BC: Analytical Applications of Differentiation
A piece of cardboard is being used to make a container that will have no lid. Four square cutouts of side length h will be cut from the corners of the cardboard. The container will have a square base of side s, height h, and a volume of 80 in^3. Which is the correct order of steps for finding minimum surface area A of the container? s^2h = 80 and A = s^2 + 4sh I. A'=2s-(320/s^2) II. 0=2s-(320.s^2) III. A=s^2+(320/s)
III, I, II
Given f'(x) = -2x^2 / sqrt(4-x^2), determine where f(x) is decreasing.
(-2,2)
The second derivative of function g is given by g''(x)=2x/(1+x^2)^2. For which interval is the graph of g concave up?
(0,inifinity)
The derivative of function f is given by f'(x)={ -2x if x<1 // x^ +2 if x>=1}. For which interval is the graph of f concave up?
(1,infinity)
Given f ′(x) = 4x^3 + 12x^2, determine the interval(s) on which f is both increasing and concave up.
(− 3, −2) and (0, infinity)
Given g(x) = 0.5x^4 + 4x, determine the x-value for the absolute minimum of the function g(x) on [−2, 3].
-cbrt(2)
Determine the x-value where a possible global maximum for a function h occurs on the interval −4 ≤ x ≤ 2 when h(x) = x^3 − 4x^2.
0
What is the global minimum of the function f(x)= |x − 2| on [−5, 5]?
0
Optimization Problems
1. draw a picture as needed 2. label any values and variables (as few variables as possible) 3. find formula, identify domain 4. rewrite formula in terms of one variable (valid substitutions) 5. use FDT and SDT to determine critical points and concavity, identify max and min 6. include units 7. does the answer make sense?
Determine Function Behavior (Curve Sketching)
1. find critical points (y'=0 or y' DNE) 2. test the regions to see if function incr or decr 3. find possible inflection points (y''=0 or y'' DNE) 4. test for concavity 5. sketch the curve (with other helpful pts)
Candidates Test (absolute extrema)
1. verify f is continuous on [a.b] 2. find all critical points on [a,b] 3. evaluate f at the critical points AND endpoints 4. identify absolute extrema (largest y-value is the abs max, smallest y-value is the abs min)
An object moves such that its velocity is defined as v(t) = −t^3 + 4t^2 + 2t for 0 ≤ t ≤ 8 seconds. When does the object reach its maximum acceleration?
1.333s
(5.01) A drone flies in a straight line with a positive velocity v(t), in miles per hour, at time t hours, where v is a differentiable function of t. Selected values of v(t) are shown in the table. t(hours) 0 1 2 3 4 5 6 v(t)(miles per hour) 2.1 2.5 2.1 1.7 1.5 1.5 1.6 Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval 0 < c < 6?
2
Find the absolute maximum of f(x)=sqrt(4-x^2) on [−2, 2].
2
Let g(x) = 3x^2 − 3x. At what value of x on the interval [0, 2] does g have a possible absolute maximum?
2
Let f be the function with derivative defined by f ′(x) = cos(x^2) on the interval −2 < x < 2. How many points of inflection does the graph of f have on this interval?
3
If f(x) = sinx, then the Mean Value Theorem guarantees that somewhere between pi/6 and pi/2, f ′(x) is equal to which value?
3/2pi
The second derivative of a function is given by f ″(x) = xcosx. How many points of inflection does f have on the interval (−2π, π)?
4
What value of b will make f(3) a local maximum if f(x) = −x^3 + bx^2 for 0 ≤ x ≤ 6?
4.5
The function f is defined by f(x) = 3x^2 − 4x + 2. The application of the Mean Value Theorem to f on the interval 0 < x < 3 guarantees the existence of a value c, where 0 < c < 3 such that f ′(c) is equal to which of the following?
5
(5.01) For what value of x does the function f(x) = x^3 − 6x^2 − 96x + 18 have a local minimum?
8
A cone-shaped paper cup is being produced such that it holds 100 cm^3 of liquid. The material that will be used to produce the cups cost 0.25 cents per cm^2. Let the cost be a function of r and the slant height of the cup be defined as s=sqrt(r^2 + h^2). Which of the following equations will help to determine the lowest cost? (Hint: the base of the cup would not be included, since it is open.)
d/dr(0.25pir*sqrt(r^2 + 90000/pi^2r^4)) =0
The function f(x) is continuous over the interval [−2, 5] and has a critical point at x = 1. If f ″ is negative on this interval, which of the following is true?
f has an absolute maximum at x = 1
Determine Concavity
f is concave up: f' is increasing: f'' > 0 (positive) f is concave down: f' is decreasing: f'' < 0 (negative)
(MVT) Mean Value Theorem
f is continuous on [a,b] differentiable on (a,b) x=a and x=b joined by secant line there must exist at least one point c between them such that the tangent line at c has the same slope as the secant line f'(c) = f(b)-f(a) / b-a
Rolle's Theorem
f is continuous on [a,b] differentiable on (a.b) f(a)=f(b) there is a number c between them such that f'(c)=0 *specific case of MVT where endpoints are equal
Extreme Value Theorem
f is continuous on [a,b] f has an absolute minimum and an absolute maximum on [a,b]
Which statement is true about a function f(x), where f'(x)=3x^2 =2/x
f is increasing for x > 0.874 because f ′(x) > 0 for x > 0.874
The table of values for f ′(x) describes the behavior of continuous function f. Which of the following is true? (1 point) x 0 1 2 3 4 f ′(x) −1 0 2 0 5
f is increasing on (1, 3) and (3, 4) because f ′(x) > 0.
Justify why the function with f ′ = 2cosx + 2cosxsinx is decreasing from pi/2 to 3pi/2
f ′(x) < 0 for (pi/2, 3pi/2)
First Derivative Test (function behavior)
f'(c)=0 --> x=c is a critical point of f(x) f'(x)>0 for all x in interval (a.b) --> f(x) is increasing on [a,b] f'(x)<0 for all x in interval (a,b) --> f(x) is decreasing on [a,b] f'(x)>0 for x<c and f'(x)<0 for x>c --> f(c) is a maximum f'(x)<0 for x<c and f'(x)>0 for x>c --> f(c) is a minimum
Second Derivative Test (concavity, local extrema)
f'(c)=0 and f''(c)<0 (conc down) --> local max at x=c f'(c)=0 and f''(c)>0 (conc up) --> local min at x=c *if f''(c)=0 SDT cannot be used
Absolute (Global) Extrema
f(c) >= f(x) for all x in the domain (max) f(c) <= f(x) for all x in the domain (min) 1. find critical points on [a,b] 2. find y value of all critical points and endpoints x=a and x=b 3. the largest y value is the abs max, the smallest y value is the abs min
Relative (Local) Extrema
f(c) >= f(x) in a certain interval (max) f(x) <= f(x) in a certain interval (min)
(5.01) The function f(x)=x^2/3 does not satisfy which of the following conditions on the interval [−1, 1] for the Mean Value Theorem to apply?
f(x) is not differentiable on (−1, 1)
Critical Point
function is not differentiable f' = 0
If a function g(x) is continuous on [a, b] and differentiable on (a, b), which of the following is not necessarily true?
g(c) = 0 for some c, such that a < c < b
Implicit Relations
if f'=0 or f' DNE --> critical point if f''>0 concave up
Graph of f, f', f''
increasing: f'(x)>0 decreasing: f'(x)<0 local min: f'(x)=0 or DNE, f''(x)>0 local max: f'(x)=0 or DNE, f''(x)<0 point of inflection: f'(x) local extrema + changes from decr to incr (vv) conc up: f''(x)>0 conc down: f''(x)<0
Point of Inflection
where the concavity changes 1. find f''(x) 2. find where f''(x)=0 or f''(x) DNE 3. check x values just to the right and left for a sign change
Given f ′(x) = 10x^4 + 16x^3, determine the x-value(s) where a point of inflection exists on the graph of f.
x=-6/5
The second derivative of the function h is given by h″(x) = 5x^3 − 15x. At what value(s) of x does the graph of h have a point of inflection?
x=-sqrt(3), x=0, x=sqrt(3)
Given f'(x) = (x^2 -8x)/(x-4)^2, determine the x-value where a relative maximum exists for f(x).
x=0
Which of the following functions cannot use the Second Derivative Test to determine if x = 3 is the location of a relative minimum or relative maximum?
y = x^3 − 9x^2 + 27x
Given y = x^3 − 2x for x ≥ 0, find the equation of the tangent line to y where the absolute value of the slope is minimized.
y=-1.089
Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0, 4]?
y=arctan(x)